Theorem lubfun 18550

Description: The LUB is a function. (Contributed by NM, 9-Sep-2018.)

Hypothesis

Ref	Expression
lubfun.u	⊢ 𝑈 = (lub‘𝐾)

Assertion

Ref	Expression
lubfun	⊢ Fun 𝑈

Proof of Theorem lubfun

Dummy variables 𝑥 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6595	. . . 4 ⊢ Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
2		funres 6599	. . . 4 ⊢ (Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) → Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))}))
3	1, 2	ax-mp 5	. . 3 ⊢ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})
4		eqid 2763	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2763	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
6		lubfun.u	. . . . 5 ⊢ 𝑈 = (lub‘𝐾)
7		biid 264	. . . . 5 ⊢ ((∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
8		id 23	. . . . 5 ⊢ (𝐾 ∈ V → 𝐾 ∈ V)
9	4, 5, 6, 7, 8	lubfval 18548	. . . 4 ⊢ (𝐾 ∈ V → 𝑈 = ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))}))
10	9	funeqd 6578	. . 3 ⊢ (𝐾 ∈ V → (Fun 𝑈 ↔ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})))
11	3, 10	mpbiri 261	. 2 ⊢ (𝐾 ∈ V → Fun 𝑈)
12		fun0 6624	. . 3 ⊢ Fun ∅
13		fvprc 6897	. . . . 5 ⊢ (¬ 𝐾 ∈ V → (lub‘𝐾) = ∅)
14	6, 13	eqtrid 2810	. . . 4 ⊢ (¬ 𝐾 ∈ V → 𝑈 = ∅)
15	14	funeqd 6578	. . 3 ⊢ (¬ 𝐾 ∈ V → (Fun 𝑈 ↔ Fun ∅))
16	12, 15	mpbiri 261	. 2 ⊢ (¬ 𝐾 ∈ V → Fun 𝑈)
17	11, 16	pm2.61i 184	1 ⊢ Fun 𝑈

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃!wreu 3367  Vcvv 3454  ∅c0 4281  𝒫 cpw 4560   class class class wbr 5106   ↦ cmpt 5189   ↾ cres 5657  Fun wfun 6549  ‘cfv 6555  ℩crio 7396  Basecbs 17413  lecple 17461  lubclub 18509

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-lub 18544

This theorem is used by:  joinfval  18571  joinfval2  18572